Adaptive Control Chapter 14: Adaptive regulation – Rejection of unknown disturbances 1 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Chapter 14: Adaptive regulation – Rejection of unknown disturbances 2 Adaptive Control – Landau, Lozano, M’Saad, Karimi
The Active Suspension System The Active Suspension System Objective: machine primary acceleration / force (disturbance) • Reject the effect of unknown main elastomere cone and variable narrow band 1 chamber piston residual acceleration (force) disturbances 4 • Do not use an aditional hole actuator controller (piston 2 measurement motor 3 position) support inertia chamber Two paths : u p (t) •Primary (disturban ce) •Secondary (double -d ⋅ q C / D differentiator) 1 Controller Plant p 1 t ( ) = T s 1 . 25 m s q -d ⋅ + u(t) + R / S B / A − y(t) (residual force) 3 Adaptive Control – Landau, Lozano, M’Saad, Karimi
The Active Suspension Elastomer Active suspension Residual force (acceleration) measurement Primary force (acceleration) (the shaker) 4 Adaptive Control – Landau, Lozano, M’Saad, Karimi
The Active Vibration Control with Inertial Actuator The Active Vibration Control with Inertial Actuator Objective: • Reject the effect of unknown and variable narrow band disturbances • Do not use an aditional measurement Same control approach but different technology Two paths : u p (t) •Primary (disturban ce) -d ⋅ •Secondary (double q C / D 1 differentiator) Controller Plant p 1 t ( ) q -d ⋅ + u(t) + R / S B / A = T s 1 . 25 m s − y(t) (residual force) 5 Adaptive Control – Landau, Lozano, M’Saad, Karimi
View of the active vibration control with inertial actuator Passive damper Residual Inertial actuator force (not visible) (acceleration) measurement Primary force (acceleration) (the shaker) 6 Adaptive Control – Landau, Lozano, M’Saad, Karimi
View of a flexible controlled structure using inertial actuators Inertial actuator Inertial actuators Accelerometers 7 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Active Suspension Frequency Characteristics of the Identified Models Secondary path Primary path = = = n 14 ; n 16 ; d 0 A B Further details can be obtained from : http://iawww.epfl.ch/News/EJC_Benchmark/ 8 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Direct Adaptive Control : disturbance rejection Disturbance : Chirp 25 Hz 47 Hz Open loop Closed loop Initialization of the adaptive controller 9 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Time Domain Results Adaptive Operation Simultaneous controller initialization Direct adaptive control and disturbance application 10 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Direct Adaptive Control Step frequency changes Output Initialization of the adaptive controller Input Adaptation transient Adaptation transient 4 Output (residual force) Output (residual force) 2 2 output 0 0 -2 20 Hz -2 32 Hz 20 Hz 32 Hz -4 29.8 29.9 30 30.1 30.2 30.3 30.4 19.8 19.9 20 20.1 20.2 20.3 20.4 20.5 20.6 Time(s) Time(s) 2 Input (control action) Input (control action) 2 1 input 0 0 -1 -2 -2 29.8 29.9 30 30.1 30.2 30.3 30.4 19.8 19.9 20 20.1 20.2 20.3 20.4 20.5 20.6 Time(s) 11 Time(s) Adaptive Control – Landau, Lozano, M’Saad, Karimi
Rejection of unknown finite band disturbances • Assumption: Plant model almost constant and known (obtained by system identification) • Problem: Attenuation of unknown and/or variable stationary disturbances without using an additional measurement • Solution: Adaptive feedback control - Estimate the model of the disturbance (indirect adaptive control) - Use the internal model principle - Use of the Youla parameterization (direct adaptive control) A robust control design should be considered assuming that the model of the disturbance is known A class of applications : suppression of unknown vibration (active vibration control) Attention: The area was “dominated “ by adaptive signal processing solutions (Widrow’s adaptive noise cancellation) which require an additional transducer Remainder : Models of stationary sinusoidal disturbances have poles on the unit circle 12 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Unknown disturbance rejection – – classical solution classical solution Unknown disturbance rejection Environment Disturbance δ ( t ) correlated Disturbance +Plant measurement model N / p D p Disturbance Plant p 1 t ( ) q -d ⋅ + u(t) + B / A FIR y(t) Obj: minimization of E{y 2 } Adaptation Algorithm Disadvantages: • requires the use of an additional transducer • difficult choice of the location of the transducer • adaptation of many parameters Not justified for the rejection of narrow band disturbances 13 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Notations S v(t) v(t) p(t) yp r(t) y(t) u(t) K G + - − − q d B ( q 1 ) − − − 1 1 1 R ( q ) R ' ( q ) H ( q ) = − G ( q 1 ) − = = 1 K ( q ) R − A ( q 1 ) − − − 1 1 1 S ( q ) S ' ( q ) H ( q ) S H R and H S are pre-specified Output Sensitivity function : − − − 1 1 1 A ( z ) S ' ( z ) H ( z ) − = 1 S ( z ) S yp − 1 P ( z ) = + Closed loop poles : − − − − − − P ( z 1 ) A ( z 1 ) S ( z 1 ) z d B ( z 1 ) R ( z 1 ) The gain of S yp is zero at the frequencies where S yp (e -j ω )=0 (perfect attenuation of a disturbance at this frequency) 14 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Disturbance model Deterministic framework − 1 N ( q ) p = ⋅ δ p ( t ) ( t ) : determinis tic disturbanc e − 1 D ( q ) p → = D poles on the unit circle; δ (t) Dirac p Stochastic framework − 1 N ( q ) = p ⋅ p ( t ) e ( t ) : stochastic disturbanc e − 1 D ( q ) p → = σ D poles on the unit circle; e(t) Gaussian w hite noise sequence (0, ) p 15 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Closed loop system. Notations Closed loop system. Notations − 1 N ( q ) p = ⋅ δ δ p ( t ) ( t ) : determinis tic disturbanc e ( t ) − 1 D ( q ) p N / p D → = D poles on the unit circle; δ (t) Dirac p p Controller : Controller p t ( ) Plant q -d ⋅ + u(t) + R / S B / A = ⋅ − − − R ( q 1 ) R ' ( q 1 ) H ( q 1 ); − R = ⋅ − − − S ( q 1 ) S ' ( q 1 ) H ( q 1 ). S Internal model principle: H S (z -1 )=D p (z -1 ) -1 -1 -1 -1 − − A(q )H (q )S' (q ) N p (q ) 1 1 1 A ( q ) S ( q ) S = ⋅ ⋅ δ = ⋅ = − ⋅ 1 y(t) ( t ) Output : y ( t ) p ( t ) S ( q ) p ( t ) - 1 - 1 − yp 1 P ( q ) P(q ) D (q ) p − = − − + − − − 1 1 1 d 1 1 CL poles : P ( q ) A ( q ) S ( q ) z B ( q ) R ( q ) 16 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Indirect adaptive regulation Indirect adaptive regulation Two-steps methodology: − 1. Estimation of the disturbance model, D p ( q 1 ) ˆ = − − H ( q 1 ) D ( q 1 ) 2. Computation of the controller, imposing S p Environment δ ( t ) Disturbance N / p D model p p t ( ) Plant Disturbance q -d ⋅ + u(t) + B / A y(t) Controller - p ˆ t ( ) Disturbance Specs. Disturbance Design Model Observer Method Estimation It can be time consuming (if the plant model B/A is of large dimension) 17 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Indirect adaptive control Indirect adaptive control Step I : Estimation of the disturbance model ARMA identification (Recursive Extended Least Squares) Step II : Computation of the controller Solving Bezout equation (for S’ and R) ˆ = H D S p ˆ − + d = A D S ' q BR P p ˆ = S D S ' p Remark: It is time consuming for large dimension of the plant model 18 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Internal model principle and Q- -parametrization parametrization) ) Internal model principle and Q Central contr: [R 0 (q -1 ),S 0 (q -1 )]. Q-parametrization : CL poles: P(q -1 )=A(q -1 )S 0 (q -1 )+q -d B(q -1 )R 0 (q -1 ). R(z 1 )=R 0 (q -1 )+A(q -1 )Q(q -1 ); S(q -1 )=S 0 (z -1 )-q -d B(q -1 )Q(q -1 ). Control: S 0 (q -1 ) u(t) = -R 0 (q -1 ) y(t) − − − n = + + 1 1 Q ( q ) q q q .... q n q Q − − 1 = − 1 Control: S ( q ) u ( t ) R ( q ) y ( t ) 0 1 Q S 0 (q -1 ) u(t) = -R 0 (q -1 ) y(t) - Q (q -1 ) w(t), The closed loop poles where w(t) = A (q -1 ) y(t) - q -d B (q -1 ) u(t). remain unchanged CL poles: P(q -1 )=A(q -1 )S 0 (q -1 )+q -d B(q -1 )R 0 (q -1 ). Plant Model Model Model = Plant w=Ap 1 19 Adaptive Control – Landau, Lozano, M’Saad, Karimi
Yula-Kucera parametrization An interpretation for the case A asympt. stable δ ( t ) N / p D p Plant p 1 t ( ) u ( t ) y ( t ) + − 1 S / q d B / A + 0 - - Model - + − q d B / A w ( t ) p 1 t ( ) Q A R 0 AN p δ = w ( t ) ( t ) D p 20 Adaptive Control – Landau, Lozano, M’Saad, Karimi
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